minimization problems and corresponding renormalized energies

MINIMIZATION PROBLEMSAND CORRESPONDING RENORMALIZED ENERGIES

Catalin LEFTER and Vicentiu RADULESCU

1. Introduction and theoretical setting

Let G be a smooth bounded simply connected domain in R2. Let a = (a1, ..., ak) be aconfiguration of distinct points in G and d = (d1, ..., dk) ∈ Zk. We also consider a smoothboundary data g : ∂G → S1 whose topological degree is d = d1 + ... + dk. Let also ρ > 0be sufficiently small and denote

Ωρ = G \k⋃

i=1

B(ai, ρ) , Ω = G \ a1, ...ak .

As in [BBH4] we consider the classes of functions

(1) Eρ,g = v ∈ H1(Ωρ; S1); v = g on ∂G and deg(v, ∂B(ai, ρ)) = di, for i = 1, ..., k

(2) Fρ = v ∈ H1(Ωρ;S1); deg(v, ∂G) = d and deg(v, ∂B(ai, ρ)) = di, for i = 1, ..., k

(3) Fρ,A = v ∈ Fρ;∫

∂G

| ∂v

∂τ|2≤ A

and the minimization problems

(4) Eρ,g = infv∈Eρ,g

∫

Ω

| ∇v |2

(5) Fρ = infv∈Fρ

∫

Ω

| ∇v |2 .

1

(6) Fρ,A = infv∈Fρ,A

∫

Ω

| ∇v |2 .

F. Bethuel, H. Brezis and F. Helein proved in [BBH4] that the minimization problems(4) and (5) have unique solutions, say uρ respectively vρ. By analysing the behaviour ofuρ as ρ → 0 they obtained the renormalized energy W (a, d, g) by the following asymptoticexpansion:

(7)12

∫

Ωρ

| ∇uρ |2= π

( k∑

i=1

d2i

)log

1ρ

+ W (a, d, g) + O(ρ) , as ρ → 0.

We shall omit d in W (a, d, g) when k = d and each dj equals 1.By considering the behavior of vρ as ρ → 0 we obtain in the first part of this paper a

notion of renormalized energy W (a, d) when only singularities and degrees are prescribed.This will appear in a similar asymptotic expansion:

(8)12

∫

Ωρ

| ∇vρ |2= π

( k∑

i=1

d2i

)log

1ρ

+ W (a, d) + O(ρ) , as ρ → 0.

The connection between the two energies is given by

(9) W (a, d) = infg:∂G→S1

deg(g,∂G)=d

W (a, d, g).

Moreover the infimum in (9) is atteint. We give thereafter an explicit formula forW (a, d).

We recall that in [BBH4] the study of the minimization problems (4) and (5) is relatedto the unique solutions Φρ respectively Φρ of the following linear problems:

(10)

∆Φρ = 0 in Ωρ

Φρ = Ci = Const. on each ∂ωi with ωi = B(ai, ρ)∫

∂ωi

∂Φρ

∂ν= 2πdi i = 1, ...k

∂Φρ

∂ν= g ∧ gτ on ∂G

∫

∂G

Φρ = 0

and

(11)

∆Φρ = 0 in Ω

Φρ = Ci = Const. on ∂ωi i = 1, ..., k

Φρ = 0 on ∂G∫

∂ωi

∂Φρ

∂ν= 2πdi i = 1, ..., k .

2

We also recall that Φρ converges uniformly as ρ → 0 to Φ0, which is the uniquesolution of

(12)

∆Φ0 = 2π

k∑

j=1

djδajin G

∂Φ0

∂ν= g ∧ gτ on ∂G

∫

∂G

Φ0 = 0 .

The explicit formula for W (a, d, g) found in [BBH4] is

(13) W (a, d, g) = −π∑

i 6=j

didj log | ai − aj | +12

∫

∂G

Φ0(g ∧ gτ )− π

k∑

i=1

diR0(ai) ,

where

R0(x) = Φ0(x)−k∑

j=1

dj log | x− aj | .

The expression we obtain for W (a, d) is lied to Φ0, which is the local uniform limit ofΦρ as ρ → 0 and is the unique solution of the problem

(14)

∆Φ0 = 2π

k∑

j=1

djδaj in G

Φ0 = 0 on ∂G .

In the second part of this section, considering the minimization problem (6) we finda variant of the formula (8), but for W replaced by WA, which is a corresponding notionof renormalized energy that satisfies

WA(a, d) = infW (a, d, g); deg (g; ∂G) = d and∫

∂G

| ∂g

∂τ|2≤ A .

In Section 3 we calculate explicitly W and W in a particular case and deduce auxiliaryresults.

In the last section we minimize the Ginzburg-Landau energy

Eε(u) =12

∫

G

| ∇u |2 +1

4ε2

∫

G

(1− | u |2)2

3

in the class

Hd,A = u ∈ H1(G;R2); | u |= 1 on ∂G, deg (u, ∂G) = d and∫

∂G

| ∂u

∂τ|2≤ A .

We prove that Hd,A is non-empty if A is big enough and the infimum of Eε is atteint.If uε is a minimizer, we prove the convergence as ε → 0 of uε to u?, which is a canon-ical harmonic map with values in S1 and d singularities, say a1, · · · , ad. Moreover, theconfiguration a = (a1, · · · , ad) minimizes the renormalized energy WA.

We recall here (see [BBH4]) that v is a canonical harmonic map with values in S1 andboundary data g if it is harmonic and satisfies

v ∧ ∂v

∂x1= −∂Φ0

∂x2in Ω

v ∧ ∂v

∂x2=

∂Φ0

∂x1in Ω ,

or, equivalently,

∂

∂x1

(v ∧ ∂v

∂x1

)+

∂

∂x2

(v ∧ ∂v

∂x2

)= 0 in D′(G) .

If v is canonical and has singularities a1, · · · , ak ∈ G with topological degrees d1, · · · , dk

then v has the form

v(z) =(

z − a1

| z − a1 |)d1

· · ·(

z − ak

| z − ak |)dk

eiϕ(z) ,

where ϕ is a smooth harmonic function in G.

2. The renormalized energy for prescribed singularities and degrees

We know from Chapter I in [BBH4] that

(15)

vρ ∧ ∂vρ

∂x1= −∂Φρ

∂x2in Ωρ

vρ ∧ ∂vρ

∂x2=

∂Φρ

∂x1in Ωρ .

So

(16) | ∇vρ |=| ∇Φρ | in Ωρ .

4

Lemma 1. Φρ converges to Φ0 in L∞(Ωρ) as ρ → 0. More precisely, there exists

C > 0 such that

(17) ‖Φρ − Φ0‖L∞(Ωρ) ≤ Cρ.

Lemma 2. Let v be a solution of

(18)

∆v = 0 in Ωρ

v = 0 on ∂G∫

∂ωj

∂v

∂ν= 0 for each j .

Then

supΩρ

v − infΩρ

v ≤k∑

j=1

(supωj

v − infωj

v) .

Proof of Lemma 2. We shall adapt the proof of Lemma I.3 in [BBH4]. Let

αj = inf∂ωj

v , βj = sup∂ωj

v and Ij = [αj , βj ] .

We shall prove for the instant that

(19)k⋃

j=1

Ij is connected .

Suppose, by contradiction, it is not true. Then, there exist t0 ∈ R , δ > 0 and 1 ≤ i ≤k such that

βj ≤ t0 − δ if 1 ≤ j ≤ i

αj ≥ t0 + δ if i + 1 ≤ j ≤ k.

We may suppose, without loss of generality that t0 6= 0, say t0 > 0. We may alsosuppose that t0 − δ ≥ 0. Choose θ ∈ C∞(R, [0, 1]) such that

θ(t) =

0 if t ≤ t0 − δ

1 if t ≥ t0 + δ .

We multiply ∆v = 0 with θ(v) and then integrate on Ωρ. Observing that θ(v) = 0 on∂G we deduce

0 =∫

Ωρ

θ′(v) | ∇v |2 −∫

∂G

∂v

∂νθ(v) +

k∑

j=1

∫

∂ωj

∂v

∂νθ(v) =

∫

Ωρ

θ′(v) | ∇v |2 .

5

So ∇v = 0 on B = x ∈ Ωρ; t0 − δ < v(x) < t0 + δ which is a contradiction.We distinguish two cases:Case 1. inf

Ωρ

v < 0 and supΩρ

v > 0.

In this case, from the connectedness ofk⋃

j=1

Ij , v = 0 on ∂G and the maximum principle,

our conclusion follows obviously.Case 2. inf

Ωρ

v = 0 or supΩρ

v = 0.

We shall treat only the first case. Suppose v 6= 0 on Ωρ (otherwise the conclusion is

obvious). By the Hopf maximum principle,∂v

∂ν< 0 on ∂G, which contradicts

∫

∂G

∂v

∂ν= 0.

Proof of Lemma 1. We apply Lemma 2 to the function v = Φρ − Φ0. SinceΦρ = Const. on each ∂B(aj , ρ), it follows that

supΩρ

(Φρ − Φ0)− infΩρ

(Φρ − Φ0) ≤k∑

j=1

(sup

∂B(aj ,ρ)

Φ0 − inf∂B(aj ,ρ)

Φ0

)≤ Cρ .

Using now the fact that Φρ − Φ0 = 0 on ∂G we obtain

(20) ‖Φρ − Φ0‖L∞(Ωρ) ≤ Cρ .

Remark. By Lemma 1 and standard elliptic estimates it follows that Φρ convergesin Ck

loc(Ω ∪ ∂G) as ρ → 0, for each k ≥ 0.

Theorem 1. As ρ → 0 then (up to a subsequence) vρ converges in Ckloc(Ω ∪ ∂G) to

v0, which is a canonical harmonic map.

Moreover, the limits of two such sequences differ by a multiplicative constant of mod-

ulus 1.

Proof. We may write, locally on Ωρ ∪ ∂G, vρ = eiϕρ with 0 ≤ ϕρ ≤ 2π. Thus, by(15),

(21)

∂ϕρ

∂x1= −∂Φρ

∂x2in Ωρ

∂ϕρ

∂x2=

∂Φρ

∂x1in Ωρ .

6

Hence, up to a subsequence, ϕρ converges in Ckloc(Ω ∪ ∂G). This means that vρ

converges (up to a subsequence) in Ckloc(Ω ∪ ∂G) to some v0. Denote by gρ = vρ|∂G. It is

clear that gρ converges to some g0 and v0 satisfies

(22)

v0 ∧ ∂v0

∂x1= −∂Φ0

∂x2in Ω

v0 ∧ ∂v0

∂x2=

∂Φ0

∂x1in Ω

v0 = g0 on ∂G ,

which means that v0 is a canonical harmonic map.We now consider two sequences vρn and vνn which converge to v1 and v2. Locally,

ϕρn → ϕ1 and ϕνn → ϕ2 .

Thus, ∇ϕ1 = ∇ϕ2, so ϕ1 and ϕ2 differ locally by an additive constant, which meansthat v1 and v2 differ locally by a multiplicative constant of modulus 1. By the connected-ness of Ω, this constant is global.

Let

R0(x) = Φ0(x)−k∑

j=1

dj log | x− aj | .

We observe that R0 is a smooth harmonic function in G.

Theorem 2. We have the following asymptotic estimate:

(23)12

∫

Ωρ

| ∇vρ |2= π

( k∑

j=1

d2j

)log

1ρ

+ W (a, d) + O(ρ) , as ρ → 0 ,

where

(24) W (a, d) = −π∑

i 6=j

didj log | ai − aj | −π

k∑

j=1

djR0(aj) .

Proof. We follow the ideas of the proof of Theorem I.7 in [BBH4].Since Φρ is harmonic in Ωρ and Φρ = 0 on ∂G we may write

12

∫

Ωρ

| ∇vρ |2= 12

∫

Ωρ

| ∇Φρ |2= −12

k∑

j=1

∫

∂B(aj ,ρ)

∂Φρ

∂νΦρ = −π

k∑

j=1

dj Φρ

(∂B(aj , ρ)

).

By Lemma 1 and the expression of R0 we easily deduce (23).

7

Theorem 3. The following equality holds:

(25) W (a, d) = infdeg (g;∂G)=d

W (a, d, g)

and the infimum is atteint.

Proof. Step 1. W (a, d) ≤ infdeg (g;∂G)=d

W (a, d, g).

Suppose not, then there exist ε > 0 and g : ∂G → S1 with deg (g; ∂G) = d such that

(26) W (a, d, g) + ε ≤ W (a, d) .

Thus, if uρ is a solution of (4), then

(27)12

∫

Ωρ

| ∇uρ |2= π

( k∑

j=1

d2j

)log

1ρ

+ W (a, d, g) + O(ρ) ≥

≥ 12

∫

Ωρ

| ∇vρ |2= π

( k∑

j=1

d2j

)log

1ρ

+ W (a, d) + O(ρ) , as ρ → 0 .

We obtain a contradiction by (26) and (27).

Step 2. If gρ and g0 are as in the proof of Theorem 1, then

W (a, d) = W (a, d, g0) .

For r > 0 let uρ,r be a solution of the minimization problem

(28) minu∈Er,gρ

∫

Ωr

| ∇u |2 .

Denote uρ,ρ = uρ and Φρ,r the solution of the associated linear problem (see (10)).Let Φρ,0 be the solution of (12) for g replaced by gρ.

We recall (see Theorem I.6 in [BBH4]) that

(29) Φρ,r → Φρ,0 in Ckloc(Ω ∪ ∂G) as r → 0

and

(30) | 12

∫

Ωr

| ∇uρ,r |2 −π

( k∑

j=1

d2j

)log

1ρ−W (a, d, gρ) |≤ Cgρr ,

where Cg = C(g) > 0 is a constant which depends on the boundary data g.

8

Our aim is to prove that Cgρis uniformly bounded for ρ > 0. Indeed, analysing the

proof of Theorem I.7 in [BBH4] we observe that Cgρ depends on Cgρ , which appears in

(31) ‖Φρ,r − Φρ,0‖L∞(Ωr) ≤k∑

j=1

[sup

∂B(aj ,r)

Φρ,0 − inf∂B(aj ,r)

Φρ,0

]≤ Cgρ

r .

It is clear at this stage, by the convergence of gρ and elliptic estimates, that Cgρis

uniformly bounded.Observe now that the map C1(∂G; S1) 3 g 7−→ W (a, d, g) is continuous. We have

| W (a, d, g0)− W (a, d) |≤| 12

∫

Ωρ

| ∇vρ |2 −π

( k∑

j=1

d2j

)log

1ρ− W (a, d) | +

+ | 12

∫

Ωρ

| ∇vρ |2 −π

( k∑

j=1

d2j

)log

1ρ−W (a, d, gρ) | + | W (a, d, gρ)−W (a, d, g0) |≤

≤ O(ρ) + Cρ+ | W (a, d, gρ)−W (a, d, g0) |→ 0 as ρ → 0 .

ThusW (a, d) = W (a, d, g0) ,

which concludes the proof of this step.

Theorem 4. For fixed A, if wρ is a solution of the minimization problem (6) then

the following holds:

(32)12

∫

Ωρ

| ∇wρ |2= π

( k∑

j=1

d2j

)log

1ρ

+ WA(a, d) + o(1) , as ρ → 0 ,

where

(33) WA(a, d) = infW (a, d, g); deg (g; ∂G) = d and

∫

∂G

| ∂g

∂τ|2≤ A ,

and the infimum is atteint.

Moreover, wρ converges in C0,α(Ω ∪ ∂G) to the canonical harmonic map associated

to g0, a, d.

Proof. The existence of wρ is obvious. Let gρ = wρ |∂G. It follows from Chapter I in[BBH4] that

(34)12

∫

Ωρ

| ∇wρ |2= π

( k∑

j=1

d2j

)log

1ρ

+ W (a, d, gρ) + Ogρ(ρ) , as ρ → 0 ,

9

where Og(η) stands for a quantity X such that | X |≤ Cgη and Cg depends only on g, a

and d.By the boundedness of gρ in H1(∂G) we may suppose that (up to a subsequence)

gρ g0 weakly in H1(∂G), as ρ → 0 .

As in the proof of Theorem 3 (see (31)) we deduce that Cgρis uniformly bounded.

We now prove that the map g 7−→ W (a, d, g) is continuous in the weak topology ofH1(∂G). Taking into account the weak convergence of gρ to g0 and the Sobolev embeddingTheorem we obtain

gρ ∧ ∂gρ

∂τ g0 ∧ ∂g0

∂τweakly in L2(∂G), as ρ → 0 .

Using (12), it follows that

Φρ,0 Φ0 weakly in H1(G), as ρ → 0 .

So, by the Rellich Theorem,

Φρ,0 → Φ0 strongly in L2(G), as ρ → 0 .

Therefore,

∫

∂G

Φρ,0

(gρ ∧ ∂gρ

∂τ

)→

∫

∂G

Φ0

(g0 ∧ ∂g0

∂τ

)as ρ → 0 .

We also deduce, using elliptic estimates, that for each i,

Rρ,0(ai) → R0(ai) as ρ → 0 .

Thus, by (13), we obtain the continuity of the map g 7−→ W (a, d, g). Hence, by (34),we easily deduce (32).

The fact that the infimum in (33) is atteint may be deduced with similar argumentsas in the proof of Theorem 3.

The convergence of wρ to a canonical harmonic map follows easily from the conver-gence of gρ.

10

3. Renormalized energies in a particular case and related properties

We shall calculate in the first part of this section the expressions of W (a, d, g) whenG = B(0; 1) and g(θ) = eidθ, for an arbitrary configuration a = (a1, ..., ak).

Proposition 1. The expression of the renormalized energy W (a, d) is given by

W (a, d) = −π∑

i 6=j

didj log | ai − aj | +π∑

i 6=j

didj log | 1− aiaj | +π

k∑

j=1

d2j log(1− | aj |2) .

Proof. If R0 is that defined in the preceding section, then

∆R0 = 0 in B1

R0(x) = −k∑

j=1

dj log | x− aj | if x ∈ ∂B1 .

It follows from the linearity of this problem that it is sufficient to calculate R0 whenthe configuration of points consists of one point, say a. Hence, by the Poisson formula, foreach x ∈ B1,

(35) R0(x) = − d

2π(1− | x |2)

∫

∂B1

log | z − a || z − x |2 dz .

We first observe that

(36) R0(x) = 0 if a = 0 .

If a 6= 0 and a? =a

| a |2 , then

(37) R0(x) = − d

2π(1− | x |2)

∫

∂B1

log | z − a? | + log | a || z − x |2 dz =

= −d log | x− a? | −d log | a | .

Hence, by (36) and (37)

(38) R0(x) =

0 if a = 0

− d log | x− a? | −d log | a | if a 6= 0 .

In the case of a general configuration a = (a1, ..., ak) one has

(39) R0(x) = −k∑

j=1

dj log | x− a?j | −

k∑

j=1

dj log | aj | .

11

It follows now from (12) that

∆(

Φ0

2π

)= dδa in B1

∂

∂ν

(Φ0

2π

)=

d

2πon ∂B1

∫

∂B1

Φ0

2π= 0 .

Thus the function Ψ =Φ0

2π− d

4π(| x |2 −1) satisfies

(44)

∆Ψ = dδa − d

πin B1

∂Ψ∂ν

= 0 on ∂B1

∫

∂B1

Ψ = 0 .

By unicity arguments, it follows from (43) and (44) that

(45)Φ0

2π− d

4π(| x |2 −1) =

d

2πlog | x− a | + 1

2πlog | x− a? | − d

4π| x |2 +C .

Taking into account the expression of C given in (42), as well as the link between Φ0

and R0 we obtain (40).

Remark. It follows by Theorem 3 and Propositions 1 and 2 that

∑

i 6=j

didj log | ai − aj | +k∑

j=1

d2j log(1− | aj |2) ≤ 0 .

A very interesting problem is the study of configurations which minimize W (a, d, g)with d and g prescribed. This relies on the behaviour of minimizers of the Ginzburg-Landauenergy (see [BBH4] for further details).

Proposition 3. If k = 2 and d1 = d2 = 1, then the minimal configuration for W is

unique (up to a rotation) and consists of two points which are symmetric with respect to

the origin.

Proof. Let a and b be two distinct points in B1. Then

−W

π= log(| a |2 + | b |2 −2 | a | · | b | · cosϕ) + log(1+ | a |2| b |2 −2 | a | · | b | · cosϕ)+

13

+ log(1− | a |2) + log(1− | b |2) ,

where ϕ denotes the angle between the vectors −→Oa and −→Ob. So, a necessary condition

for the minimum of W is cos ϕ = −1, that is the points a, O and b are colinear, with O

between a and b. Hence one may suppose that the points a and b lie on the real axis and−1 < b < 0 < a < 1. Denote

f(a, b) = 2 log(a− b) + 2 log(1− ab) + log(1− a2) + log(1− b2) .

A straightforward calculation, based on the Jensen inequality and the symmetry of f ,shows that a = −b = 5−1/4.

4. The behavior of minimizers of the Ginzburg-Landau energy

We assume throughout this section that G is strictly starshaped about the origin.In [BBH2] and [BBH4], F. Bethuel, H. Brezis and F. Helein studied the behavior of

minimizers of the Ginzburg-Landau energy Eε in

H1g (G;R2) = u ∈ H1(G;R2); u = g on ∂G ,

for some smooth fixed g : ∂G → S1, deg (g; ∂G) = d > 0. Our aim is to study a similarproblem, that is the behavior of minimizers uε of Eε in the class

(46) Hd,A = u ∈ H1(G;R2); | u |= 1 on ∂G, deg (u, ∂G) = d and∫

∂G

| ∂u

∂τ|2≤ A .

It would have seemed more naturally to minimize Eε in the class

Hd = u ∈ H1(G;R2); | u |= 1 on ∂G, deg (u, ∂G) = d

but, as observed by F. Bethuel, H. Brezis and F. Helein, the infimum of Eε is not atteint.To show this, they considered the particular case when G = B1, d = 1 and g(x) = x. Thisis the reason which imposed us to take the infimum of Eε on the class Hd,A, that was alsoconsidered by F. Bethuel, H. Brezis and F. Helein.

Theorem 5. For each sequence εn → 0, there is a subsequence (also denoted by εn)

and exactly d points a1, · · · , ad in G such that

uεn → u? in H1loc(G \ a1, · · · , ad;R2) ,

where u? is a canonical harmonic map with values in S1 and singularities a1, · · · , ad of

degrees +1.

14

Moreover, the configuration a = (a1, · · · , ad) is a minimum point of

WA(a, d) := min W (a, d, g); deg (g; ∂G) = d and

∫

∂G

| ∂g

∂τ|2≤ A .

Proof. Step 1. The existence of uε.For fixed ε, let un

ε be a minimizing sequence for Eε in Hd,A. It follows that (up to asubsequence)

unε uε weakly in H1

and, by the boundedness of unε |∂G in H1(∂G), we obtain that

uεn |∂G→ uε |∂G strongly in H12 (∂G) .

This means that, if gε = uε |∂G, then

deg (gε; ∂G) = d .

By the lower semi-continuity of Eε, uε is a minimizer of Eε. Moreover, this uε satisfiesthe Ginzburg-Landau energy

(47) −∆uε =1ε2

uε(1− | uε |2) in G .

Step 2. A fundamental estimate.As in the proof of Theorem III.2 in [BBH4], multiplying (47) by x·∇uε and integrating

on G, we find

(48)12

∫

∂G

(x · ν)(

∂uε

∂ν

)2

+1

2ε2

∫

G

(1− | uε |2)2 =

=12

∫

∂G

(x · ν)(

∂gε

∂τ

)2

−∫

∂G

(x · τ)∂uε

∂ν

∂gε

∂τ.

Using now the boundedness of gε in H1(∂G) and the fact that G is strictly starshapedwe easily obtain

(49)∫

∂G

| ∂uε

∂ν|2 +

1ε2

∫

G

(1− | uε |2)2 ≤ C ,

where C depends only on A and d.

Step 3. A fundamental Lemma.The following result is an adapted version of Theorem III.3 in [BBH4] which is essential

towards locating the singularities at the limit.

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Lemma 3. There exist positive constants λ0 and µ0 (which depend only on G, d and

A) such that if1ε2

∫

G∩B2`

(1− | uε |2)2 ≤ µ0 ,

where B2` is some disk of radius 2` in R2 with

`

ε≥ λ0 and ` ≤ 1 ,

then

(50) | uε(x) |≥ 12

if x ∈ G ∩B` .

The proof of Lemma is essentially the same as of the cited theorem, after observingthat

‖∇uε‖L∞(G) ≤C

ε,

where C depends only on G, d and A.

Step 4. The convergence of uε.Using Lemma 1 and the estimate (49), we may apply the methods developed in Chap-

ters III-V in [BBH4] to determine the “bad” disks, as well as the fact that their numberis uniformly bounded. The same techniques allow us to prove the weak convergence inH1

loc(G \ a1, · · · , ak;R2) of a subsequence, also denoted by uεn , to some u?.As in [BBH4], Chapter X (see also [S]) one may prove that, for each p < 2,

uεn → u? in W 1,p(G) .

This allows us to pass at the limit in

∂

∂x1

(uεn ∧

∂uεn

∂x1

)+

∂

∂x2

(uεn ∧

∂uεn

∂x2

)= 0 in D′(G)

and to deduce that u? is a canonical harmonic map.The strong convergence of (uεn) in H1

loc (G \ a1, · · · , ak;R2) follows as in [BBH4],Theorem VI.1 with the techniques from [BBH3], Theorem 2, Step 1.

We then observe that for all j, deg (u?, aj) 6= 0. Indeed, if not, then as in Step1 of Theorem 2 in [BBH3], the H1-convergence is extended up to aj , which becomes a“removable singularity”. The fact that all these degrees equal +1 and the points a1, · · · , ad

are not on the boundary may be deduced as in Theorem VI.2 [BBH4].The following steps are devoted to characterize the limit configuration as a minimum

point of the renormalized energy WA.

Step 5. An upper bound for Eε(uε).

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For R > 0, let I(R) be the infimum of Eε on H1g (G) with G = B(0; ε

R ) and g(x) =x

| x |on ∂G. Following the ideas of the proof of Lemma VIII.1 in [BBH4] one may show thatif b = (bj) is an arbitrary configuration of d distinct points in G and g is such that

deg (g, ∂G) = d and∫

∂G

| ∂g

∂τ|2≤ A, then there exists η0 > 0 such that, for each η < η0,

(51) Eε(uε) ≤ dI

(ε

η

)+ W (b, g) + πd log

1η

+ O(η) , as η → 0

for ε > 0 small enough. Here O(η) stands for a quantity which is bounded by Cη, whereC is a constant depending only on g.

Step 6. A lower bound for Eεn(uεn).With the same proof as of Step 2 of Theorem 1 in [LR] one may show that if a1, · · · , ad

are the singularities of u? and η > 0, then there is N0 = N0(η) ∈ N such that, for eachn ≥ N0,

(52) Eεn(uεn) ≥ dI

(εn

η(1 + η)

)+ πd log

1η

+ W (a, g0) + O(η) ,

where O(η) is a quantity bounded by Cη, where C depends only on g0.

Step 7. The limit configuration is a minimum point for WA.Taking into account that (see [BBH4], Chapter III)

I(ε) = π | log ε | +γ + O(ε) ,

we obtain by (51) and (52)

(53) W (b, g)− πd log εn + dγ + O

(εn

η

)≥

≥ W (a, g0)− πd log εn + dγ + O(η) .

Adding πd log εn in (53) and passing to the limit firstly as n →∞ and then as η → 0,we find

(54) W (a, g0) ≤ W (b, g) .

As b and g are arbitrary chosen it follows that a = (a1, · · · , ad) is a global minimumpoint of

(55) WA(b) = min W (b, g); deg (g; ∂G) = d and∫

∂G

| ∂g

∂τ|2≤ A .

Remark. The infimum in (55) is atteint because of the continuity of the mappingHd,A 3 g 7−→ W (b, g) with respect to the weak topology of H1(∂G).

Acknowledgements. We are grateful to Prof. H. Brezis for his constant supportduring the preparation of this work. We also thank Th. Cazenave for useful discussions.

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REFERENCES

[BBH1] F. Bethuel, H. Brezis and F. Helein, Limite singuliere pour la minimisationdes fonctionnelles du type Ginzburg-Landau, C.R. Acad. Sc. Paris 314(1992), 891-895.

[BBH2] F. Bethuel, H. Brezis and F. Helein, Tourbillons de Ginzburg-Landau etenergie renormalisee, C.R. Acad. Sc. Paris 317(1993), 165-171.

[BBH3] F. Bethuel, H. Brezis and F. Helein, Asymptotics for the minimization of aGinzburg-Landau functional, Calculus of Variations and PDE, 1(1993), 123-148.

[BBH4] F. Bethuel, H. Brezis and F. Helein, Ginzburg-Landau Vortices, Birkhauser,1994.

[LR] C. Lefter and V. Radulescu, On the Ginzburg-Landau energy with weight, toappear.

[S] M. Struwe, On the asymptotic behavior of minimizers of the Ginzburg-Landaumodel in 2 dimensions, Diff. Int. Eq., 7(1994), 1613-1624.

C. Lefter: Current address: Laboratoire d’Analyse Numerique, Tour 55-65Universite Pierre et Marie Curie4, place Jussieu75252 Paris Cedex 05, FRANCE.Permanent address: Department of MathematicsUniversity of Iasi6600 Iasi, ROMANIA.

V. Radulescu: Current address: Laboratoire d’Analyse Numerique, Tour 55-65Universite Pierre et Marie Curie4, place Jussieu75252 Paris Cedex 05, FRANCE.Permanent address: Department of MathematicsUniversity of Craiova1100 Craiova, ROMANIA.

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minimization problems and corresponding renormalized energies

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